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Mathematical Structures

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The webpages collected here list information about classes of mathematical structures. The aim is to have a central place to check what properties are known about these structures.

Initially the main content concerns first-order classes of relational structures and, more particularly, varieties of universal algebras. If you are familiar with some of these classes of algebras, feel free to add some relevant information and references by using the edit link at the bottom of the respective page. A phrase followed by a blue '?' is a link to a page that still needs to be filled in. Adding and formatting pages is really simple. Here are some basic [text formatting rules] (see also Wikipedia[how to edit a page], although not all the features mentioned there are available here). Mathematical notation is delimited by dollar signs, and uses a very basic subset of LaTeX (essentially those characters and symbols that can be displayed by most browsers). You can use this sandbox page to try out editing (without worrying about deleting useful information).

New: There is now an experimental XHTML+MathML version of these pages at http://math.chapman.edu/cgi-bin/structuresxml.pl (go to [Display MathML on these pages] to check if your browser can handle MathML formulas).

Suggestions or comments? Just follow the link and edit the page.

Please use this template for new pages and follow the format of other pages that have already been partially filled out. If possible, the value of a property should be justified by a brief reason or a reference to a publication where the result can be found. Bibtex references (e.g. from MathSciNet) can be automatically translated to the bibxml format using the [bibtex2xml converter].

This front page can also be edited to add further classes of structures. A criterion for inclusion in this list is that there should be some journal publication or book where the class has been named and defined.

Acknowledgements | Notation and terminology | Properties | Tools | Online books and lecture notes

Varieties | Quasivarieties | Universal classes | First-order classes | Second-order classes

Classes of ordered algebraic structures | Partial algebras | Many-sorted algebras | Logics

Alphabetical list of all classes

  1. Abelian groups
  2. Abelian lattice-ordered groups
  3. Action algebras
  4. Action lattices
  5. Algebraic lattices
  6. Algebraic posets
  7. Algebraic semilattices
  8. Allegories?
  9. Almost distributive lattices
  10. [Associative algebras]
  11. [Banach spaces]
  12. Bands
  13. Basic logic algebras
  14. BCI-algebras
  15. BCK-algebras
  16. BCK-join-semilattices
  17. BCK-lattices
  18. BCK-meet-semilattices
  19. BL-algebras
  20. Boolean algebras
  21. Boolean algebras with operators
  22. Boolean groups
  23. Boolean lattices
  24. Boolean modules over a relation algebra
  25. Boolean monoids
  26. Boolean rings
  27. Boolean semigroups
  28. Boolean semilattices
  29. Boolean spaces
  30. Bounded distributive lattices
  31. Bounded lattices
  32. [Bounded residuated lattices]?
  33. Brouwerian algebras
  34. Brouwerian semilattices
  35. [C-star-algebras]
  36. Cancellative commutative monoids
  37. Cancellative commutative semigroups
  38. Cancellative monoids
  39. Cancellative semigroups
  40. Cancellative residuated lattices
  41. [Categories]
  42. Chains
  43. Clifford semigroups
  44. [Clifford algebras]
  45. Closure algebras
  46. Commutative BCK-algebras
  47. Commutative groupoids
  48. Commutative inverse semigroups
  49. Commutative monoids
  50. Commutative regular rings
  51. Commutative residuated lattices
  52. Commutative rings
  53. Commutative rings with identity
  54. Commutative semigroups
  55. [Compact topological spaces]
  56. [Compact zero-dimensional Hausdorff spaces]?
  57. Complemented lattices
  58. Complemented distributive lattices
  59. Complemented modular lattices
  60. [Complete distributive lattices]?
  61. Complete lattices
  62. Complete semilattices
  63. Complete partial orders
  64. [Completely regular Hausdorff spaces]?
  65. Completely regular semigroups
  66. [Continuous lattices]?
  67. [Continuous posets]?
  68. [Cylindric algebras]?
  69. De Morgan algebras
  70. [De Morgan monoids]?
  71. [Dedekind categories]?
  72. [Dedekind domains]?
  73. Dense linear orders
  74. [Digraph algebras]?
  75. Directed complete partial orders
  76. Directed partial orders
  77. [Directed graphs]?
  78. Directoids
  79. [Distributive allegories]?
  80. Distributive double p-algebras
  81. Distributive dual p-algebras
  82. Distributive lattices
  83. [Distributive lattices with operations]?
  84. [Distributive lattice ordered semigroups]?
  85. Distributive p-algebras
  86. Distributive residuated lattices
  87. [Division algebras]?
  88. Division rings
  89. Double Stone algebras
  90. [Dunn monoids]?
  91. [Dynamic algebras]?
  92. Entropic groupoids
  93. [Equivalence algebras]?
  94. Euclidean domains
  95. f-rings
  96. Fields
  97. FL-algebras
  98. FLc-algebras?
  99. FLe-algebras
  100. FLew-algebras?
  101. FLw-algebras?
  102. Frames?
  103. Function rings
  104. G-sets?
  105. Generalized BL-algebras
  106. Generalized Boolean algebras
  107. Generalized MV-algebras
  108. Goedel algebras
  109. Graphs?
  110. Groupoids
  111. Groups
  112. [Halmos algebras]?
  113. Hausdorff spaces
  114. Heyting algebras
  115. [Hilbert algebras]?
  116. Hilbert spaces
  117. Hoops
  118. Idempotent semirings
  119. Idempotent semirings with identity
  120. Idempotent semirings with identity and zero
  121. Idempotent semirings with zero
  122. Implicative lattices
  123. Integral domains
  124. [Integral relation algebras]?
  125. [Integral residuated lattices]?
  126. [Intuitionistic linear logic algebras]?
  127. Inverse semigroups
  128. Involutive lattices
  129. [Involutive residuated lattices]?
  130. Join-semidistributive lattices
  131. Join-semilattices
  132. [Jordan algebras]?
  133. Kleene algebras
  134. Kleene lattices
  135. [Lambek algebras]?
  136. Lattice-ordered groups
  137. Lattice-ordered rings
  138. Lattices
  139. Left cancellative semigroups
  140. [Lie algebras]?
  141. Linear Heyting algebras
  142. [Linear logic algebras]?
  143. Linear orders
  144. Locales?
  145. [Locally compact topological spaces]
  146. Loops
  147. Lukasiewicz algebras of order n
  148. M-sets?
  149. Medial groupoids
  150. [Medial quasigroups]?
  151. Meet-semidistributive lattices
  152. Meet-semilattices
  153. Metric spaces
  154. Modal algebras
  155. Modular lattices
  156. [Modular ortholattices]?
  157. Modules over a ring
  158. [Monadic algebras]?
  159. [Monoidal t-norm logic algebras]?
  160. Monoids
  161. Moufang loops
  162. [Moufang quasigroups]?
  163. [Multiplicative additive linear logic algebras]?
  164. [Multiplicative lattices]?
  165. Multisets?
  166. MV-algebras
  167. Neardistributive lattices
  168. Near-rings
  169. Near-rings with identity
  170. Near-fields
  171. [Nilpotent groups]?
  172. Nonassociative relation algebras
  173. Normal bands
  174. [Normed vector spaces]?
  175. Ockham algebras
  176. Order algebras
  177. [Ordered fields]?
  178. [Ordered rings]?
  179. Ordered sets
  180. [Ore domains]?
  181. Ortholattices
  182. Orthomodular lattices
  183. p-groups?
  184. [Partial groupoids]?
  185. Partially ordered sets
  186. [Peirce algebras]?
  187. Pocrims?
  188. Polrims?
  189. Posets
  190. Pointed residuated lattices
  191. [Post algebras]?
  192. Preordered sets
  193. [Priestley spaces]?
  194. Principal Ideal Domains
  195. [Process algebras]?
  196. [Pseudo basic logic algebras]?
  197. [Pseudo MTL-algebras]?
  198. [Pseudo MV-algebras]?
  199. Pseudocomplemented distributive lattices
  200. [Pure discriminator algebras]?
  201. Quantales?
  202. Quasigroups
  203. Quasi-ordered sets
  204. Rectangular bands
  205. Regular rings
  206. Regular semigroups
  207. Relation algebras
  208. [Relative Stone algebras]?
  209. [Relativized relation algebras]?
  210. [Representable cylindric algebras]?
  211. [Representable relation algebras]?
  212. Residuated lattices
  213. [Residuated idempotent semirings]?
  214. Rings
  215. Rings with identity
  216. [Schroeder categories]?
  217. [Semiassociative relation algebras]?
  218. Semidistributive lattices
  219. Semigroups
  220. Semigroups with identity
  221. Semigroups with zero
  222. Semilattices
  223. Semilattices with identity
  224. Semilattices with zero
  225. Semirings
  226. Semirings with identity
  227. Semirings with identity and zero
  228. Semirings with zero
  229. Sequential algebras
  230. Sets
  231. Shells
  232. Skew-fields
  233. [Small categories]?
  234. [Sober T0-spaces]?
  235. [Solvable groups]?
  236. [Stably compact spaces]?
  237. [Steiner quasigroups]?
  238. Stone algebras
  239. T0-spaces
  240. T1-spaces
  241. T2-spaces
  242. [Tarski algebras]?
  243. Tense algebras
  244. [Temporal algebras]?
  245. [Topological groups]?
  246. Topological spaces
  247. [Topological vector spaces]
  248. [Torsion groups]?
  249. Trees?
  250. Tournaments?
  251. [Unary algebras]?
  252. Unique factorization domains
  253. Unital rings
  254. Vector spaces over a field
  255. Wajsberg algebras
  256. Wajsberg hoops
  257. [Weakly associative lattices]?
  258. [Weakly associative relation algebras]?
  259. [Weakly representable relation algebras]?

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Last edited November 11, 2004 10:55 pm (diff)
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